131-133] Van der Waals' Equation 115 



Van der Waals' Equation. 



133. According to Van der Waals, equation (269), 



pv = RNT (275), 



in which v is now written for the volume instead of fl, to agree with the 

 usual notation in this subject, must be corrected in two ways. The first 

 correction is a correction to be applied to the term v to represent the finite 

 size of the molecules, and the second is a correction to be applied to the 

 term p, to represent the influence upon the pressure of the forces of cohesion 

 in the gas. 



The argument of Van der Waals as to the first correction is as follows*. 

 Let there be N molecules each of diameter cr, and let us suppose the centre 

 of each surrounded by a sphere of radius <r, and therefore of volume f7r<r 3 . 

 In considering possible positions for the centre of molecule A, we know that 

 it cannot lie within any of the N 1 spheres surrounding the N I other 

 molecules, so that the space available for the centre of A must not be taken 

 to be v but v (N 1)1^0^. This expression, it is true, requires correction 

 on account of the possibility of two or more of the N1. spheres over- 

 lapping, but this correction will be of a higher order of small quantities than 

 that already made, and may therefore be neglected. 



Also the expression requires correction owing to the impossibility of the 

 centre of a sphere being within a distance ^ of the boundary. This correc- 



M 



tion requires us further to reduce v to the extent of the volume of a layer of 



thickness taken round the boundary of the containing vessel, but clearly 



x 



this correction may be neglected if a- vanishes in comparison with the 

 dimensions of the vessel. This condition is, of course, entirely different 

 from the condition that the sums of the volumes of the molecules shall be 

 small compared with the volume of the vessel. The former condition is 

 satisfied if o-y~* may be neglected, the latter is satisfied if Na 3 /v can be 

 neglected. Using the figures given in 8, and taking the case of a gas at 

 atmospheric pressure in a vessel of 1 litre capacity, we find 



<nr* = 2 x 10-, 

 N(T S /V = 3-2 x 10~ 4 . 



It is therefore rational to neglect the one correction, while taking the other 

 into account. 



* As regards method of presentation, I have followed Boiczmann (Gastheorie, n. p. 7) more 

 closely than the original work of Van der Waals, 



82 



